If you increase the steepness of the ramp, then you will increase the
acceleration of a ball which rolls down the ramp. This can be seen in
two different ways:
1) Components of forces. Forces are vectors and have a direction
and a magnitude. The force of gravity points straight down, but a ball
rolling down a ramp doesn't go straight down, it follows the ramp.
Therefore, only the component of the gravitational force which points
along the direction of the ball's motion can accelerate the ball. The
other component pushes the ball into the ramp, and the ramp pushes
back, so there is no acceleration of the ball into the ramp. If the
ramp is horizontal, then the ball does not accelerate, as gravity
pushes the ball into the ramp and not along the surface of the ramp. If
the ramp is vertical, the ball just drops with acceleration due to
gravity. These arguments are changed a bit by the fact that the ball is
rolling and not sliding, but that only affects the magnitude of the
acceleration but not the fact that it increases with ramp steepness.
2) Work and energy. The change in potential energy of the ball is
its mass times the change in height (only the vertical component counts
-- horizontal displacements do not change gravitational potential
energy) times the local gravitational acceleration g. This loss of
gravitational potential energy shows up as an increase in kinetic
energy. If the ball falls a farther distance vertically, it will have a
greater kinetic energy and be going faster. Again, the kinetic energy
is shared between the motion of the ball going somewhere, and the
rotation of the ball, and so the details of the acceleration depend on
the ball (is it hollow or solid?), but the dependence on the steepness
of the ramp is the same.
Tom
(published on 10/22/2007)
